With two-electrode configuration the specific capacitance of one

With two-electrode configuration the specific capacitance of one electrode can be calculated by using the voltammetric charge integrated from a CV curve by Equation 1: 𝑉

4

C = 𝑚∆𝑉𝑣 ∫𝑉 + 𝐼(𝑉)𝑑𝑉

(1)

−

where m is the total mass of the active materials in both electrodes, 𝑣 is the scan rate, and ∆𝑉 is the potential window. The multiplier of 4 adjusts the cell capacitance and the total mass to the capacitance and the mass of a single electrode. If area or volume is more important, then m can be substituted by the electrode area or volume). The specific capacitance in one electrode is often calculated from a galvanostatic charge– discharge curve by using Equation 2: 4𝐼∆𝑡

C = 𝑚∆𝑉

(2)

where ∆𝑡 is the discharging time, I is the current applied on the electrodes, m is the total mass of active materials in both electrodes, and ∆𝑉 is the potential window of the discharging process. ∆𝑉/∆𝑡 is actually the slope of the discharge curve. As most commercial supercapacitors operate in the range of Vmax to ~½Vmax, these two voltage values are recommended for the determination of the slope [1297]. It is worth noting that the cell capacitance calculation by Galvanostatic discharge has been established in the supercapacitor industry, which correlates more closely to the electric loading and unloading in the majority of applications. At a constant scan rate, the average specific power density during discharge can be calculated by integrating the CV curves using Equation 11: 1

𝑉

P = 𝑚𝑉 ∫0 + 𝐼V𝑑𝑉

(11)

where V is the potential difference and m is the total mass of two electrodes. And the average specific energy density can be calculated by using Equation 12: 1

𝑉

E = 3600𝑚𝑣 ∫0 + 𝐼V𝑑𝑉

(12)

where m is the total mass of two electrodes and 𝑣 is the scan rate. Moreover, the maximum energy density and power density for two-electrode configuration can be calculated by Equation 13 and 14, respectively: 1

𝐸2 = 2 𝐶2 𝑉 2 𝑉2

𝑃2 = 4𝑅

2

(13) (14)

where 𝐶2 is the specific capacitance calculated from two-electrode configuration, V is the potential window, and 𝑅2 is the equivalent series resistance for two-electrode configuration. It is clear that a larger potential window V has positive effect on both energy density and power density. And energy density is proportional to the specific capacitance. From Equation (14), low internal resistance is very important for achieving high power density so that high conductivity of active materials is essential for high power density.